Theoretical Preliminaries of Acoustics
In this chapter, we review some of the fundamentals of acoustics and introduce the spherical harmonic expansion of a sound field, which is the basis for the spherical harmonic processing framework used with spherical microphone arrays. This material provi
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Theoretical Preliminaries of Acoustics
In this chapter, we review some of the fundamentals of acoustics and introduce the spherical harmonic expansion of a sound field, which is the basis for the spherical harmonic processing framework used with spherical microphone arrays. This chapter intends to introduce the key theory and equations required in the rest of the book. For a more comprehensive introduction to acoustics, the reader is referred to [2, 12], or [17, 20] for a thorough treatment of acoustics in spherical coordinates.
2.1 Fundamentals of Acoustics The propagation of acoustic waves through a material is described by a second-order partial differential equation known as the wave equation. The homogeneous wave equation describes the evolution of the sound pressure p as a function of time t and position r = (x, y, z) in a homogeneous, source-free medium.1 In three dimensions it is given by [12, Eq. 1.5] ∇ 2 p(r, t) − where ∇2 =
1 ∂ 2 p(r, t) = 0, c2 ∂t 2
∂2 ∂2 ∂2 + + ∂x 2 ∂y2 ∂z2
(2.1)
(2.2)
this section, vectors in Cartesian coordinates are denoted with a corner mark to distinguish them from vectors in spherical coordinates, which will be introduced in Sect. 2.2.
1 In
© Springer International Publishing Switzerland 2017 D. Jarrett et al., Theory and Applications of Spherical Microphone Array Processing, Springer Topics in Signal Processing 9, DOI 10.1007/978-3-319-42211-4_2
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2 Theoretical Preliminaries of Acoustics
is the Laplace operator in Cartesian coordinates (x, y, z) and c denotes the speed of sound. The separation of variables method is used to simplify the analysis. The time-harmonic solution to the wave equation can then be written in the form p(r, t) = P(r, k)eiωt ,
(2.3)
√ where i = −1, and P(r, k), to be defined later in this section, is a function of the position r and the wavenumber ffl k. The wavenumber is related to the angular frequency ω, ordinary frequency and speed of sound c via the dispersion relation ffl 2π ω . k= = c c
(2.4)
The acoustic waves are assumed to be propagating in a non-dispersive medium, such that the propagation speed c is independent of the wavenumber k. Throughout this book, the speed of sound is assumed to be constant; when a numerical value is required, we will use c = 343 m/s, obtained when the medium is air at a temperature of approximately 19 ◦ C [12, Eq. 1.1]. The function P(r, k)eiωt in (2.3) can be represented in the complex plane by a rotating vector or a phasor. The time-independent vector, represented by the complex number P(r, k), is the complex amplitude. The complex amplitude is multiplied by the unit vector eiωt rotating anti-clockwise at speed ω (in rad · s−1 ), which is the angular frequency of the harmonic function. Warning: Throughout this book, eiωt represents the time dependence of a positivefrequency wave; a convention that is commonly adopted in electrical and mechanical engineering. In Sect. 2.3, we will summarize the effect of the choice of convention on the key equations of this chapter. The Fourier transform of
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